for F (a set of functional dependencies on a relation scheme) is a set of dependencies such that F logically implies all dependencies in
logically implies all dependencies in F. The set
has two important properties: A canonical cover is not unique for a given set of functional dependencies, therefore one set F can have multiple covers
In the following example, Fc is the canonical cover of F. Given the following, we can find the canonical cover: R = (A, B, C, G, H, I), F = {A→BC, B→C, A→B, AB→C} Fc = {A → B, B →C} An attribute is extraneous in a functional dependency if its removal from that functional dependency does not alter the closure of any attributes.
[2] Given a set of functional dependencies
and a functional dependency
and any of the functional dependencies in
{\displaystyle (F-(A\rightarrow B)\cup {(A-a)\rightarrow B})}
using Armstrong's Axioms.
[2] Using an alternate method, given the set of functional dependencies
, and a functional dependency X → A in
, attribute Y is extraneous in X if
[3] For example: Given a set of functional dependencies
and a functional dependency
{\displaystyle (F-(A\rightarrow B)\cup \{A\rightarrow (B-a)\})}
using Armstrong's axioms.
[3] A dependent attribute of a functional dependency is extraneous if we can remove it without changing the closure of the set of determinant attributes in that functional dependency.